Finite Group Databasebeta

← #366918

Subgroups — #366918

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5#20647C211yesmaximal, characteristic
6#20441C211yesmaximal, characteristic
7C2×C4×#101C211yesmaximal, characteristic
8#11366C212yesmaximal
9#11128C212yesmaximal
10#10942C212yesmaximal
11#10869C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#475—24no—
15C2×#475—21no—
16C2×#475—21no—
17C2×#475—22no—
18C2×#475C418yescomplemented
19C22×#108C2212yes—
20C22×#106C2212yes—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419C2212yes—
24C22×#103C2212yes—
25C22×#101C2211yescharacteristic
26C22×#101C2211yescharacteristic
27C2×#405C2212yes—
28C2×C4×M16C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2285C2211yescharacteristic
31#1997—24no—
32#1993—22no—
33#1990—22no—
34#1950—24no—
35#1805—22no—
36#1803—22no—
37#1771—22no—
38#1770—21no—
39C8×C4⋊C4—21no—
40C4×#101—24no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2211yescharacteristic
44C2×#343C2211yescharacteristic
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49#1729C2212yes—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C23×C8C2312yes—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2×C414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8C2×C414yes—
74C2×#117C2312yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#475—28no—
82#475—28no—
83#475—216no—
84C2×#108—24no—
85C2×#108—24no—
86C2×#108—42no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—24no—
90C2×#106C2×C418yes—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C414yes—
96C2×C4⋊C8D414yes—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D412yes—
100#419—24no—
101#419—42no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103C2×C418yes—
107C2×#101—22no—
108C2×#101—22no—
109C2×#101—22no—
110C2×#101—22no—
111C2×#101—22no—
112C2×#101Q814yescomplemented
113#405—24no—
114#405—42no—
115C4×M16—22no—
116C4×M16—22no—
117C4×M16—22no—
118C4×M16D414yes—
119C2×#97—21no—
120C2×#97—22no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—22no—
124C2×C4×C8—21no—
125#384D412yes—
126#382D412yes—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131#343—24no—
132#343—24no—
133#343—22no—
134#343—22no—
135#341—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#336—22no—
140#336—22no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—42no—
156C22×Q8—84no—
157C22×Q8C22×C412yes—
158C22×Q8C2×Q812yes—
159C22×Q8#4112yes—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4#4111yescharacteristic
164C23×C4#4111yescharacteristic
165C23×C4C2411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16C2×D412yes—
191C2×M16C2×D412yes—
192C2×M16C2×Q812yes—
193C2×M16C2×Q812yes—
194C2×M16#4112yes—
195C2×M16#4112yes—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8C22×C414yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D414yes—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C22⋊C4—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—42no—
258C2×C42C22×C412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42#4112yes—
266#108—48no—
267#106—28no—
268#106—28no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8C22⋊C414yes—
272C4⋊C8C22⋊C414yes—
273C4⋊C8C22⋊C418yes—
274#103—28no—
275#103—28no—
276#103—48no—
277#101—28no—
278#101—28no—
279#97—24no—
280#97—44no—
281#97—42no—
282C4×C8—24no—
283C4×C8—44no—
284C4×C8—42no—
285#95—22no—
286#95—22no—
287C24C2×#4111yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—88no—
318C2×Q8—84no—
319C2×Q8—88no—
320C2×Q8—84no—
321C2×Q8C4×Q814yes—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—82no—
339C22×C4—84no—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12412yes—
347C22×C4#12412yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361C22×C42−1+411yescharacteristic
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—42no—
378M16—42no—
379M16—42no—
380M16—42no—
381M16C22⋊Q814yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8C2×C22⋊C414yes—
406C2×C8C2×C22⋊C414yes—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—42no—
426C42—42no—
427C42—44no—
428C42—44no—
429C42C2×C22⋊C412yes—
430C42C2×C22⋊C412yes—
431C42C4×D412yes—
432C42C4×D412yes—
433C42#12112yes—
434C42#12112yes—
435C42#12712yes—
436C42#12712yes—
437C23—22no—
438C23—22no—
439C23C2×C4×D411yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×#12411yescharacteristic
442C23#52911yescharacteristic
443C23#53311yescharacteristic
444C23#54111yescharacteristic
445C23#54211yescharacteristic
446Q8—24no—
447Q8—24no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—88no—
455Q8—88no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—84no—
481C2×C4#39012yes—
482C2×C4#39012yes—
483C2×C4#39012yes—
484C2×C4#39012yes—
485C2×C4#47114yes—
486C2×C4C22×C22⋊C411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×Q811yescharacteristic
489C2×C4C4×#4111yescharacteristic
490C2×C4#51911yescharacteristic
491C2×C4#52011yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4#52511yescharacteristic
495C2×C4C2×#12711yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53711yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54611yescharacteristic
500C2×C4#54611yescharacteristic
501C2×C4#54711yescharacteristic
502C2×C4#54711yescharacteristic
503C2×C4#54711yescharacteristic
504C2×C4#54811yescharacteristic
505C2×C4#54811yescharacteristic
506C2×C4#54811yescharacteristic
507C2×C4D4×Q811yescharacteristic
508C2×C4D4×Q811yescharacteristic
509C2×C4D4×Q811yescharacteristic
510C2×C4#55211yescharacteristic
511C2×C4#55211yescharacteristic
512C2×C4#55411yescharacteristic
513C2×C4#55611yescharacteristic
514C8—24no—
515C8—24no—
516C8—24no—
517C8—42no—
518C8—42no—
519C8—42no—
520C8—82no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—21no—
528C22—21no—
529C22C2×#39011yescharacteristic
530C22C4×C22⋊Q811yescharacteristic
531C22#232311yescharacteristic
532C22#242711yescharacteristic
533C22#242911yescharacteristic
534C22#248911yescharacteristic
535C22#249211yescharacteristic
536C22C2×#47112yes—
537C22#306912yes—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—44no—
542C4—82no—
543C4—84no—
544C4—84no—
545C4#197714yes—
546C4#233911yescharacteristic
547C4Q8×C22⋊C411yescharacteristic
548C4#236011yescharacteristic
549C4#236211yescharacteristic
550C4#241511yescharacteristic
551C4#241711yescharacteristic
552C4#254011yescharacteristic
553C4#254211yescharacteristic
554C2—21no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2#1399711yescharacteristic
559C2C2×#197711yescharacteristic
560C2#2125011yescharacteristic
561C2#2536112yes—
562C2#2549212yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
18. C2×#475C4semidirect
112. C2×#101Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4752
C2×#19972
#213242
C2×#19502
#206472
#204412
C2×C4×#1012
#113662
#111282
#109422
#108692